-<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Integration on an annulus · FastTransforms.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="FastTransforms.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit">FastTransforms.jl</span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../dev/">Development</a></li><li><span class="tocitem">Examples</span><ul><li class="is-active"><a class="tocitem" href>Integration on an annulus</a></li><li><a class="tocitem" href="../automaticdifferentiation/">Automatic differentiation through spherical harmonic transforms</a></li><li><a class="tocitem" href="../chebyshev/">Chebyshev transform</a></li><li><a class="tocitem" href="../disk/">Holomorphic integration on the unit disk</a></li><li><a class="tocitem" href="../halfrange/">Half-range Chebyshev polynomials</a></li><li><a class="tocitem" href="../nonlocaldiffusion/">Nonlocal diffusion on <span>$\mathbb{S}^2$</span></a></li><li><a class="tocitem" href="../padua/">Padua transform</a></li><li><a class="tocitem" href="../semiclassical/">Semi-classical Jacobi polynomials</a></li><li><a class="tocitem" href="../sphere/">Spherical harmonic addition theorem</a></li><li><a class="tocitem" href="../spinweighted/">Spin-weighted spherical harmonics</a></li><li><a class="tocitem" href="../subspaceangles/">Subspace angles</a></li><li><a class="tocitem" href="../triangle/">Calculus on the reference triangle</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Integration on an annulus</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Integration on an annulus</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaApproximation/FastTransforms.jl/blob/master/examples/annulus.jl" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Integration-on-an-annulus-1"><a class="docs-heading-anchor" href="#Integration-on-an-annulus-1">Integration on an annulus</a><a class="docs-heading-anchor-permalink" href="#Integration-on-an-annulus-1" title="Permalink"></a></h1><p>In this example, we explore integration of the function:</p><div>\[ f(x,y) = \frac{x^3}{x^2+y^2-\frac{1}{4}},\]</div><p>over the annulus defined by <span>$\{(r,\theta) : \rho < r < 1, 0 < \theta < 2\pi\}$</span> with parameter <span>$\rho = \frac{2}{3}$</span>. We will calculate the integral:</p><div>\[ \int_0^{2\pi}\int_{\frac{2}{3}}^1 f(r\cos\theta,r\sin\theta)^2r{\rm\,d}r{\rm\,d}\theta,\]</div><p>by analyzing the function in an annulus polynomial series. We analyze the function on an <span>$N\times M$</span> tensor product grid defined by:</p><div>\[\begin{aligned}
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